Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV

Abstract

We obtain asymptotics of polynomials satisfying the orthogonality relations $${\int }_{\mathbb{R}}{z}^k{P}_n(z;t,N){\mathrm{e}}^{-N\left(\frac{1}{4}{z}^4+\frac{t}{2}{z}^2\right)}\mathrm{d}z=0\text{ for }k=0,1,...,n-1,$$ where the complex parameter $t$ is in the so-called two-cut region. As an application, we deduce asymptotic formulas for certain families of solutions of Painlevé-IV which are indexed by a non-negative integer and can be written in terms of parabolic cylinder functions. The proofs are based on the characterization of orthogonal polynomials in terms of a Riemann-Hilbert problem and the Deift-Zhou non-linear steepest descent method.

Updated version: 42 pages, 6 figures. The focus of the manuscript was revised and content rearranged. The title and abstract were updated accordingly. Various typos/errors were corrected